Material selection

Material selection is a step in the process of designing any physical object. In the context of product design, the main goal of material selection is to minimize cost while meeting product performance goals.^[1] Systematic selection of the best material for a given application begins with properties and costs of candidate materials. For example, a thermal blanket must have poor thermal conductivity in order to minimize heat transfer for a given temperature difference.

Systematic selection for applications requiring multiple criteria is more complex. For example, a rod which should be stiff and light requires a material with high Young's modulus and low density. If the rod will be pulled in tension, the specific modulus, or modulus divided by density $E/\rho$, will determine the best material. But because a plate's bending stiffness scales as its thickness cubed, the best material for a stiff and light plate is determined by the cube root of stiffness divided by density $\sqrt[3 ]{E}/\rho$. For a stiff beam in bending the material index is $\sqrt[2 ]{E}/\rho$.

Ashby plots

Plot of Young modulus vs density. The colors represent families of materials.

An Ashby plot, named for Michael Ashby of Cambridge University, is a scatter plot which displays two or more properties of many materials or classes of materials.^[2] These plots are useful to compare the ratio between different properties. For the example of the stiff/light part discussed above would have Young's modulus on one axis and density on the other axis, with one data point on the graph for each candidate material. On such a plot, it is easy to find not only the material with the highest stiffness, or that with the lowest density, but that with the best ratio $E/\rho$. Using a log scale on both axes facilitates selection of the material with the best plate stiffness $\sqrt[3]{E}/\rho$.

Plot of Young modulus vs density with log-log scaling. The colors represent families of materials.

The first plot on the right shows density and Young's modulus, in a linear scale. The second plot shows the same materials attributes in a log-log scale. Materials families (polymers, foams, metals, etc.) are identified by colors.^[3]

Thus as energy prices have increased and technology has improved, automobiles have substituted increasing amounts of light weight magnesium and aluminium alloys for steel, aircraft are substituting carbon fiber reinforced plastic and titanium alloys for aluminium, and satellites have long been made out of exotic composite materials.

Of course, cost per kg is not the only important factor in material selection. An important concept is 'cost per unit of function'. For example, if the key design objective was the stiffness of a plate of the material, as described in the introductory paragraph above, then the designer would need a material with the optimal combination of density, Young's modulus, and price. Optimizing complex combinations of technical and price properties is a hard process to achieve manually, so rational material selection software is an important tool.

Example

A common method for choosing an appropriate material is an “Ashby chart”. By plotting a performance index for a specific case of loading on the Ashby chart, a material with maximum performance can be selected. The performance index takes into consideration the dimensional constraints, material constraints, and free variable constraints of a specific application. The following example will show the how to come up with the performance index and how to plot and interpret the Ashby chart.

This example will take into consideration a beam that will undergo two different loads with the goal of minimizing weight. The first load is a beam in tension. Figure 1 illustrates this loading.

Figure 1. Beam under Tensile stress loading to minimize weight.

The parameters for the beam can be organized into categories. These categories are material variables, which include density, modulus, and yield stress, free variables which are variables that can change during the loading cycle, for example applied force. The final category is design variables which usually are a limit of how thick the beam can be, how much it can deflect, or any other limiting factor for the specific application.

For this loading cycle, the stress in the beam is measured as $P/A$, where $P$ is the load and $A$ is the cross sectional area. The weight is measure as $w = \rho AL$, where $\rho$ is the density, and $L$ is the length. By looking at the equation, we see that for a fixed length of $L$, the material variables are $\sigma$ and $\rho$. There is one free variable, $A$, and a variable that needs to be minimized, $w$.

In order to find the performance index, an equation for w in terms of fixed and material variables needs to be found. This means that the variable A has to somehow be replaced. By rearranging the axial stress equation, $A$ can be represented as $A = P/\sigma$. Substituting this into the weight equation, $w=\rho P/\sigma L$, gives an equation for weight that has only fixed and material variables.

The next step is to separate the material variables from all other variables and constants. The equation becomes $w = (\rho/\sigma)LP$. Since the goal is to minimize weight, the material variables have to be minimized. This means that $(\rho/\sigma)$ has to be minimized, or the inverse equation, $(\sigma/\rho)$ has to be maximized. We call the equation that needs to be maximized our performance index. $P_{cr}=(\rho/\sigma)$. It is important to note that the performance index is always an equation that needs to be maximized, so inverting an equation that needs to be minimized is necessary.

The performance index can then be plotted on the Ashby chart by converting the equation to a log scale. This is done by taking the log of both sides, and plotting it similar to a line with $P_{cr}$ being the y-axis intercept. This means that the higher the intercept, the higher the performance of the material. By moving the line up the Ashby chart, the performance index gets higher. Each materials the line passes through, has the performance index listed on the y-axis. So, moving to the top of the chart while still touching a region of material is where the highest performance will be.

The next loading cycle will have a different performance index with a different equation. For example, if it is desired to maximize this beam for bending, using the max tensile stress equation of bending $\sigma = (-My)/I$, where $M$ is the bending moment, $y$ is the distance from the neutral axis, and $I$ is the moment of inertia. This is shown in Figure 2. Using the weight equation above and solving for the free variables, the solution arrived at is $w=\sqrt{6MbL^2}(\rho/\sqrt{\sigma})$, where $L$ is the length and $b$ is the height of the beam. This turns the material performance index into $P_{CR}=\sqrt{\sigma}/\rho$.

Figure 2. Beam under bending stress. Trying to minimize weight

By plotting the two performance indices on the same Ashby chart, the maximum performance index of both loading types together will be at the intercept of the two lines. This is shown in figure 3

Figure 3. Ashby chart with performance indices plotted for maximum result

As seen from figure 3 the two lines intercept near the top of the graph at Technical ceramics and Composites. This will give a performance index of 120 for tensile loading and 15 for bending. When taking into consideration the cost of the engineering ceramics, especially because the intercept is around the Boron carbide, this would not be the optimal case. A better case with lower performance index but more cost effective solutions is around the Engineering Composites near CFRP.

References

1. George E. Dieter (1997). "Overview of the Materials Selection Process", ASM Handbook Volume 20: Materials Selection and Design.
2. Ashby, Michael (1999). Materials Selection in Mechanical Design (3rd ed.). Burlington, Massachusetts: Butterworth-Heinemann. ISBN 0-7506-4357-9.
3. Ashby, Michael F. (2005). Materials Selection in Mechanical Design. USA: Elsevier Ltd. p. 251. ISBN 978-0-7506-6168-3.

External links

• Cambridge University - Profile of Prof. Michael Ashby